Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110914632815 -0.83 0.80 7.0 3.48s
2 -1.117208123450 -2.20 -1.92 0.80 1.0 1.62s
3 -1.117250534410 -4.37 -2.72 0.80 1.0 19.7ms
4 -1.117251159666 -6.20 -3.51 0.80 1.0 19.4ms
5 -1.117251188028 -7.55 -4.12 0.80 1.0 19.7ms
6 -1.117251190058 -8.69 -5.26 0.80 1.0 32.7ms
7 -1.117251190142 -10.07 -5.67 0.80 2.0 22.6ms
8 -1.117251190142 + -12.66 -5.62 0.80 1.0 20.8ms
9 -1.117251190143 -12.05 -7.01 0.80 1.0 26.1ms
10 -1.117251190143 -13.83 -7.38 0.80 2.0 23.0ms
11 -1.117251190143 -14.51 -8.30 0.80 1.0 110ms
12 -1.117251190143 + -14.57 -8.83 0.80 2.0 32.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.36 0.80 1.0 3.70s
2 -1.117251190143 + -14.51 -10.36 0.80 1.0 21.7ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 20.9s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997979 -2.46 0.80 1.0 13.7ms
2 -1.117520586053 -5.59 -3.49 0.80 1.0 20.4ms
3 -1.117520612986 -7.57 -4.23 0.80 1.0 22.3ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 77.2ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.9ms
6 -1.117520613465 -12.15 -6.42 0.80 1.0 20.3ms
7 -1.117520613465 -13.32 -7.40 0.80 1.0 20.6ms
8 -1.117520613465 -14.70 -8.21 0.80 1.0 22.9ms
9 -1.117520613465 + -14.95 -8.93 0.80 1.0 39.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264592 -1.69 0.80 1.0 13.2ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 18.7ms
3 -1.118340011112 -6.01 -3.44 0.80 1.0 19.2ms
4 -1.118340028553 -7.76 -4.36 0.80 1.0 25.4ms
5 -1.118340029137 -9.23 -5.12 0.80 1.0 19.5ms
6 -1.118340029164 -10.57 -5.62 0.80 1.0 19.8ms
7 -1.118340029166 -11.73 -6.68 0.80 1.0 25.5ms
8 -1.118340029166 -13.54 -7.58 0.80 1.0 21.0ms
9 -1.118340029166 -14.81 -8.12 0.80 1.0 21.0ms
10 -1.118340029166 + -14.48 -8.55 0.80 1.0 23.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -9.05 0.80 1.0 13.1ms
2 -1.118340029166 -15.35 -10.36 0.80 1.0 24.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029166 -11.05 0.80 1.0 13.1ms
2 -1.118340029166 -14.88 -11.42 0.80 1.0 25.1ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029166 │ │ 0.00297055 │ 3.07s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721506 -3.09 0.80 1.0 19.3ms
2 -1.118346864758 -6.84 -4.11 0.80 1.0 19.0ms
3 -1.118346866235 -8.83 -4.86 0.80 1.0 24.6ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 19.5ms
5 -1.118346866262 -12.08 -6.55 0.80 1.0 19.6ms
6 -1.118346866262 -13.42 -7.04 0.80 1.0 22.7ms
7 -1.118346866262 -15.05 -8.13 0.80 1.0 20.6ms
8 -1.118346866262 -14.88 -8.95 0.80 1.0 22.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.2ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.6ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 21.3ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 19.4ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 22.0ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 20.0ms
7 -1.118356101386 -13.58 -7.65 0.80 1.0 22.8ms
8 -1.118356101386 -14.57 -8.45 0.80 1.0 20.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.1ms
2 -1.118356101386 -14.15 -9.82 0.80 1.0 21.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 15.9ms
2 -1.118356101386 + -15.18 -11.46 0.80 1.0 18.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 534ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.2ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 24.8ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 19.1ms
4 -1.118356102877 -14.12 -7.62 0.80 1.0 19.4ms
5 -1.118356102877 + -14.75 -8.39 0.80 1.0 85.8ms
6 -1.118356102877 + -15.05 -8.89 0.80 1.0 20.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.5ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 19.0ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 19.0ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 19.4ms
5 -1.118356105120 -14.65 -7.86 0.80 1.0 19.9ms
6 -1.118356105120 + -14.88 -8.35 0.80 1.0 20.3ms
7 -1.118356105120 -15.65 -9.45 0.80 1.0 21.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.8ms
2 -1.118356105120 -14.45 -11.02 0.80 1.0 20.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 14.9ms
2 -1.118356105120 -14.51 -12.44 0.80 1.0 19.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.09557e-8 │ 1.09s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -1.36191e-34, 1.72478e-34]u"a₀")
Atom(H, [ 1.44318, -1.85624e-34, -2.6504e-35]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.48636586486918 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335324546536 -0.81 0.80 7.7 780ms
2 -8.340903061565 -2.25 -1.69 0.80 1.3 639ms
3 -8.341207818736 -3.52 -2.74 0.80 2.7 36.1ms
4 -8.341214252717 -5.19 -3.38 0.80 3.7 44.2ms
5 -8.341214271157 -7.73 -4.30 0.80 1.3 33.4ms
6 -8.341214272463 -8.88 -5.73 0.80 2.3 68.3ms
7 -8.341214272517 -10.27 -5.78 0.80 4.3 46.2ms
8 -8.341214272518 -11.90 -6.91 0.80 1.0 27.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.46 0.80 1.0 624ms
2 -8.341214272518 -13.97 -7.64 0.80 1.0 26.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.35 0.80 1.0 22.8ms
2 -8.341214272518 + -14.15 -8.49 0.80 1.0 26.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 1.64346e-33 │ 0.163458 │ -0.16 │ 30.2s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350579035748 -1.21 0.80 8.7 72.6ms
2 -8.351864454619 -2.89 -1.88 0.80 1.3 29.9ms
3 -8.351927013749 -4.20 -2.79 0.80 3.3 40.8ms
4 -8.351928508140 -5.83 -3.65 0.80 3.7 45.0ms
5 -8.351928514382 -8.20 -4.66 0.80 2.0 56.0ms
6 -8.351928514742 -9.44 -5.36 0.80 4.3 45.5ms
7 -8.351928514754 -10.92 -6.38 0.80 4.3 44.4ms
8 -8.351928514755 -12.67 -7.18 0.80 3.7 41.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353381570619 -1.52 0.80 8.0 70.4ms
2 -8.353729086677 -3.46 -2.21 0.80 1.0 26.6ms
3 -8.353757343139 -4.55 -2.95 0.80 4.0 42.7ms
4 -8.353757519138 -6.75 -4.15 0.80 2.3 32.8ms
5 -8.353757531583 -7.90 -4.89 0.80 3.7 45.8ms
6 -8.353757531643 -10.22 -5.55 0.80 2.7 35.8ms
7 -8.353757531647 -11.42 -6.83 0.80 2.7 35.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531647 -7.54 0.80 1.0 23.0ms
2 -8.353757531647 -14.05 -8.11 0.80 1.0 26.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531647 -8.72 0.80 1.0 22.9ms
2 -8.353757531647 + -Inf -8.90 0.80 1.0 26.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531647 -9.39 0.80 1.0 23.4ms
2 -8.353757531647 + -14.45 -9.49 0.80 1.0 26.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531647 │ │ │ 0.0156413 │ -0.016 │ 6.48s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353720355278 -3.20 0.80 8.3 65.1ms
2 -8.353760403803 -4.40 -3.91 0.80 1.0 27.3ms
3 -8.353770532641 -4.99 -3.86 0.80 6.0 55.9ms
4 -8.353770536531 -8.41 -4.16 0.80 1.0 26.8ms
5 -8.353770536960 -9.37 -4.51 0.80 2.3 32.3ms
6 -8.353770537318 -9.45 -5.48 0.80 2.0 32.1ms
7 -8.353770537330 -10.91 -5.97 0.80 2.7 36.7ms
8 -8.353770537331 -12.57 -6.51 0.80 3.3 37.2ms
9 -8.353770537330 + -13.57 -6.41 0.80 2.3 35.7ms
10 -8.353770537331 -13.30 -7.36 0.80 1.0 27.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353774626694 -2.00 0.80 8.3 66.0ms
2 -8.353876729535 -3.99 -2.70 0.80 1.0 26.4ms
3 -8.353895058683 -4.74 -3.33 0.80 4.0 44.3ms
4 -8.353895106332 -7.32 -4.23 0.80 2.0 46.4ms
5 -8.353895110718 -8.36 -5.08 0.80 3.3 41.1ms
6 -8.353895110768 -10.31 -5.75 0.80 3.3 40.5ms
7 -8.353895110770 -11.73 -7.60 0.80 2.0 34.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -7.93 0.80 1.0 23.4ms
2 -8.353895110770 -14.45 -8.49 0.80 1.0 27.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -8.86 0.80 1.0 23.4ms
2 -8.353895110770 -14.75 -9.07 0.80 1.0 26.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110770 -9.36 0.80 1.0 24.0ms
2 -8.353895110770 -14.75 -9.55 0.80 1.0 27.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110770 │ -3.86 │ │ 0.00064618 │ -0.00065 │ 2.85s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353778794683 -3.44 0.80 7.3 61.4ms
2 -8.353875605599 -4.01 -3.63 0.80 1.0 30.9ms
3 -8.353895211685 -4.71 -3.78 0.80 6.0 57.3ms
4 -8.353895211742 -10.24 -3.95 0.80 1.0 27.2ms
5 -8.353895212184 -9.36 -4.33 0.80 1.0 27.3ms
6 -8.353895212430 -9.61 -4.67 0.80 1.3 32.5ms
7 -8.353895212511 -10.09 -5.49 0.80 2.0 32.7ms
8 -8.353895212513 -11.85 -5.34 0.80 3.3 40.9ms
9 -8.353895212516 -11.54 -5.77 0.80 1.0 31.2ms
10 -8.353895212516 -12.19 -6.53 0.80 1.7 31.6ms
11 -8.353895212516 -13.37 -6.94 0.80 3.0 39.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353796567727 -3.40 0.80 8.0 65.1ms
2 -8.353878620163 -4.09 -3.85 0.80 1.0 27.1ms
3 -8.353895362458 -4.78 -4.22 0.80 6.0 62.4ms
4 -8.353895362649 -9.72 -4.76 0.80 1.3 29.2ms
5 -8.353895362682 -10.48 -5.27 0.80 1.3 29.7ms
6 -8.353895362685 -11.44 -5.77 0.80 1.7 31.1ms
7 -8.353895362686 -12.49 -7.33 0.80 2.0 36.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.20 0.80 1.0 23.7ms
2 -8.353895362686 -14.15 -8.21 0.80 1.0 32.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.88 0.80 1.0 24.0ms
2 -8.353895362686 -14.75 -8.86 0.80 1.0 26.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.45 0.80 1.0 23.2ms
2 -8.353895362686 + -14.75 -9.42 0.80 1.0 26.2ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71105e-6 │ -1.7e-6 │ 3.02s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [2.22552e-67 5.27783 5.27783;
5.27783 -2.22552e-67 5.27783;
5.27783 5.27783 2.22552e-67]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662944162885 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.